How to find the determinant of this matrix?

[LCSphysicist]

Homework Statement

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Relevant Equations

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I think you all can see that $a_{(i+1,j+1)} = a_{i,j} + a_{i+1,j} + a_{i,j+1}$

Now the determinant always give me problem. I have and idea to reduce this matrix by Chio to a 2x2 matrix and find the determinant of this 2x2.

Put i was not able to see any pattern to find what how the 2x2 matrix would be (beside symmetric)

Any tips?

 

[fresh_42]

I would first try to write the matrix as a product of two simpler matrices, because the construction rule is similar to matrix multiplication. If this would be too complicated, I'd try the polynomial method: $\det = \sum_{\sigma\in S_n}(-1)^n \ldots$

 

[docnet]

I want to suggest using row operations to reduce the matrix to something more manageable.

What could be helpful is the following.

Adding or subtracting any two rows of a matrix does not change the determinant.
Exchanging two rows of a matrix changes the sign of the determinant.

 

[fresh_42]

Another idea is to prove by induction that the determinant equals $2^{(n^2-n)/2}=2^{\binom n 2}$.